The distance between a system and the set of uncontrollable systems

systems

Citation for published version (APA):

Eising, R. (1982). The distance between a system and the set of uncontrollable systems. (Memorandum COSOR; Vol. 8219). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1982

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne

Take down policy

If you believe that this document breaches copyright please contact us at:

openaccess@tue.nl

providing details and we will investigate your claim.

Memorandum COSOR 82 - 19

The distance between a system and the set of uncontrollable systems

by

Rikus ~ng

Eindhoven, the Netherlands November 1982

by

Rikus Eising

Abstract. A controllability measure Ln terms of the distance between a system and the set of uncontrollable systems LS developed. Some properties of a minimal disturbance, rendering a system noncontrollable, are given.

Keywords: controllability, controllability measure, distance between a system and the set of uncontrollable systems, disturbance.

I. Introduction

In [ 1 ] Paige argues that the traditional methods for testing the

controll-, l' f ( ) nXn nxm , f

abl. l.ty 0 a system A,B where A € lR ,B € lR are not satl.S actory

in the sense thay they may provide the wrong answer and furthermore they do

agt give au answer to the question: !lis a system well/badly controllable?"

Using unitary state space transformations one can obtain a block Hessenberg

form for a system (A,B). This Hessenberg form allows one to give a reliable

answer to the ques tion concerning controllability of (A, B). However, this

algorithm only provides a "yes/no answer';. From a practical point of view it is more important to have some quantitative answer to the issue of controllability of a system in the sense that one would like to know how close a given system is to an uncontrollable system. This is important

when-ever (A,B) has been obtained on the basis of measurements.

Paige proposes in [ 1 ] to use "distance to an uncontrollable system" as a controllability measure. In fact he proposes "minimum" (oA,oB)11

2 such that (A + oA, B + oB) is uncontrollable" as such a measure ("·"2 is the spectral norm).

This paper is also concerned with controllability measures. We will provide an answer to

(I) IIWbat is the distance between a system (A,B) and the set of

uncontrollable systems?"

Of course the set of uncontrollable systems consists of systems with the

The distance b between (A,B) and (A + oA, B + oB) is defined as

The weighting factors d

A and dB' both positive, have been introduced, in order to be able to deal with cases where for instance the confidence in the A-measurements differs from the confidence in the B-measurements. We use the Frobenius norm II • II : II M 112 := trace Ml\i for a (possibly complex)

matrix M. Here'MH denotes the conjugate transpose of M. In the first

part of this paper the Frobenius norm coincides with the spectral norm:

II M II; == maximum eigenvalue of

h.

We will use the following characterization of controllability of (A,B)

(3) rank[AI - A,B]

=

n for all A ~ a: .

2. Main results

The answer to (I) is provided by the solution of the following minimization

problem (for a given system (A,B».

with respect .. to. (oA,oB) and such that (A + oA, B + oB) LS uncontrollable.

The minimum distance in (4) will denoted as d{(A,B),UNCO} (here UNCO denotes

the set of uncontrollable systems). We use A2 instead of

~

because

~2

is

easier to handle.

Diiect"m:i.iiIiiilzation of -(4) is almost always impossible so we have to exploit the information which is in the uncontrollability of (A + oA, B + oB) for any disturbance (oA,oB) of the system.~,B). I f (A + oA, B + oB) is not controllable then there exists a nonzero, possibly complex, row vector xH

such that

H

Here x denotes the conjugate transpose of a vector x and A is an

eigen-value of

A

+

cA.

We will now take a different point of view with respect to (5):

Given

A

€ JRnxn , B € JRnxm , x € (Cn, x

:f

O. Determine

cA

E (Cnxtl, oB £ (Cnxm

such that (5) is satisfied for some A E (C.

Observe that we allow complex disturbances. here. Later on we will deal with the strictly real case.

The fact that we will allow complex disturbances may seem to be unattractive

but in a number of cases, for instance if we use our measure of controll~ .

ability as a condition number for pole assignability, this may not be

unreason-n nxn

able. Let x €

a: ,

A € JR • Then we have

(6) XHA

=

-H- x xHAx H + (H x A - -H- x xHAx H) = )I. H

x X

xx xx

H H H

Observe that x and xb are orthogonal (we use x y as the inner product (x,y) in (Cn). Because we want (5) to hold we have to satisfy

(7)

=

' xH _{AX + Xo} H _{+ x} H~A v

=

,H AX

for some A E

a:.

Thus we must have

The smallest oA (II oAU2 minimal), satisfying (8), is (see [2 J)

with

H H H

oA

=

x(-x _o + px ) / x x

Because we still may choose p we take p

=

a

in order to minimize ileA

112.

The disturbance SB on B has to be taken such that

H

x (B + oB)

=

a •

Therefore the smallest tiB (II SB 112 minimal), satisfying (5), is

with

Here

~_T

stands for transposition. Now we have obtained

nxn nxm n

THEOREM. Le t A € lR ,B E lR , x € IC , x '" 0, d A > a , dB > O. Then the (possibly'complex) disturbance (&A,oB) such that

(i) x (A H + 8A)

=

AX H for some A € IC, x H (B + oB)

=

0

is given by

( H xHAx H\ H

oA

=

x -x A + -H- x ) / x x x x

the minimal value in (ii) is

(9)

and

PROOF. Using the arguments above and the fact that we may minimize

lIoAII2 and lIoBII2 separately, because both d

A and dB are positive, the proof

is complete.

0

H

From now on we will suppose that x x

=

1 because oA and oB do not depend on II x II (as was to be expected).

Using this theorem it is immediately clear how to compute the distance d{(A,B),UNCa} between (A,B) and the set of uncontrollable systems.

(10) d{ (A,B) ,UNCO} 2

=

minimum d H H T H T

A x ACI - xx )A x + dB x B B x x€(Cn,1I x 11=1

Observe that the gradient and the Hessian of the object function in (10) can easily be derived. This is advantageous for the actual computation of

d{ (A, B), UNCa}. Of course we can also compute d { (A, B) ,UNCO} using uncon-strained minimization of (9). However, this gives rise to a number of pro-blems because (9) does not depend on II x II. (Up to now the spectral norm and the Frobenius norm coincide because the disturbances turned out to be rank-one matrices.)

Next we consider the case of real disturbances (oA,oB). We also start with (8)

H H H

Let x ::: xr + ix., Xo

=

x. + ix . where x , x., x , x . are real vectors.

J. or oJ. r J. or oJ.

Let p ::: a + i~. Because we want

oA

to be real we must have

(11 )

r

ax;

+

"xq

=

TTl

L

~xr - aX_i

J

r ...

T

--x or ""T

L

X . oJ.

Using the Moore Penrose inverse we obtain as minimum norm solution for

oA

(the Moore Penrose inverse of a matrix M is denoted as M+)

I

xT .... -+

r

_~

-'r or

(12)

oA

:::

bxd

""T _{x .} oJ.

Straightforward calculation of

lloAU2

gives

(13)

T -T - T -T,... T ... T ....

2 X.X.X x J. J. or or - 2x x.x.x r J. oJ. or + x x x .x . r r oJ. oJ.

II ;SA II = --=--...;..",,---=-~=----..;;...---T T T T x x x.x. - x x.x.x r r J. J. r J. J. r ~ T T T whenever x x x.x. - x x.x.x

=

det ~ O. r r J. J. r J. l. r

Observe that if det , 0

T -x. l. = [x ,-x. ] r l. T xx r r T -x.X J. r T -x x. _r l. T x.x. l. 1. -)

If det ::: 0 then xr and Xi are dependent. Suppose that xr

=

tx

i , Then we also

have x ::: tx .• In order t'o be able to satisfy (8) with a real ;SA we must

or oJ.

(14) (15) 11 oAlI 2

=

xT x / xTx + p 2 or or r r T 2 x.x. + p ~ ~ (x :f 0) r (x. :f 0) • ~

If x. :f 0 and x :f 0 and xr

=

tx. (tE~) then (14) and (15) are the same

~ r ~

because x

=

tx .•

or o~

We still have to choose p in (8) such that

oA

becomes a minimal distu~bance

satisfying

A

Ea:.

This ~s obtained by taking p

=

0 in (14) and (15) and by minimizing (13) with respect to.; cr and ]..1.

A straightforward calculation of cr and ]..1, minimizing (13), gives

T T T T T T T T x.x.x x + x x x .x. - x x.x.x - x x.x .x ~ ~ or r r r o~ ~ r ~ ~ or r ~ o~ r cr

=

~---~~~~~--~~~~---~~--2 det (16) T T T T T T T T X~X.x. x. + x x.x x - x x.x .x. - x x x .x ~ ~ or ~ r ~ or r r ~ O~ ~ r r O~ r ]..1 = ---=---~----~---~----~~~---(x:x.)2 + (xTx )2 + 2(xTx.)2 ~ ~ r r r ~

for the disturbance

oB

we have

B .

The minimum norm solution for

oB

is

-1+

_T T . xr

I

-x. r oB

=

B T T -x. _xiJ ~

with T T T T T T 2 x.x.x BB x - 2x x.x.BB x T T T + x x x.BB x. (17) II oB II == 1. 1. r r r 1. 1. r r r 1. 1. whenever det ~ O.

The analogues of (14) and (IS} are

(18) (19) T xx r r T x. x. . 1. 1. det T T Now we could minimize (for x x

r r + X.X. l. 1. == 1)

in order to find a minimal real disturbance (oA,oB) where II 6A 112 is gl.ven by (13), (14) on (15) depending upon det being zero or not. In (13) we have

2 to substitute (16) and in (14) and (15) we have to take p == O. For II oB II

we take (17), (18) or (19) whichever. appropriate.

2

However, II 6A II is not a continuous function when xr and x. tend to dependency.

1.

This can be seen as follows. Suppose that x. ~ 0 (x ~ 0 can be handled

1. r

T

analogously). Let xr == t~i + Ep where p ~ 0 and xiP = O. Then we let £ tend

to zero~ and; (13) together with (16) generally will not tend to (15) with p

=

O. This can be seen as follows.

T T T

We substitute x == tx. + Ep in (11) and (16) and we obtain r 1. Here -T x or U ::: a + (J, V

=

a

₊ ~, _{a +}

is

= ~ xHAx xx ""'T T = tx . + Eq 01.

Now (13) becomes Using T l~oA 112 = q q + T P P -T~ x .x . o~ Ol. T X.x. ~ l. T

limu=~+

e:-l>() 2pTp T x.Ax. l. l. T 2x.x. l. 1. lim y

_"i;,\

e:+O € (I+t )x.x. l. l. TA x.Ax. T _{l. l.}

lim;(J

=.L:e.-

_{T T} -= (JO

e:+O 2p p 2x.x. we have 1l.m • ....T x. (>

=

e:+O Ol. Therefore we obtain -T x.A -l. 1. l. T x.Ax. T 1. l. T X i x.x. 1. l. T x .X.,. Ol. Dl. T X.X. l. l.

Thereby proving the possible discontinuity of II oAI12 i f xr an xi tend to dependency.

I t is easily seen that II oB 112 is a continuous function for xr and xi tending to dependency.

Because 8A is not continuous we have that the noncontrollable e~genvalue

of (A + oA,B + oB) is also not continuous. This can be seen as follows. For (oA,oB) computed as above (based on x) we have

T T T T

In the open region x x x.x. - x.x x x. f 0 u + iv is the noncontrollable r r ~ ~ 1 r r ~

eigenvalue. If xr and xi tend T

for x

=

tx. we have x.Ax. /

r ~ 1 1

to dependency u + iv tends to U

_o

whereas T

x.x. as the noncontrollable eigenvalue. ~ ~

This possible discontinuity in the, to be minimized, object function (20) may present serious problems.

Therefore (10) is to be preferred whenever possible.

3. Comparison of real and complex controllability measures

Up to now we have obtained two controllability measures characterized by

(c) complex disturbances are allowed (r) only real disturbances are allowed.

Obviously, we have that case (c) generally gives a smaller controllability

measure than case (r) (the distance between a system (A,B) and the set of

uncontrollable systems UNCO, measured using only real disturbances, gene-rally is larger than the distance between (A,B) and UNCO measured in terms of complex disturbances). The difference between case (c) and case (r) may

be investigated as follows. The vectors xr and x. may be taken to be ~

orthogonal because instead of the vector x in (5) we may take e icpx • Such a factor e icp does not affect (the norm of) the dis turbances oA and

oB. Furthermore it can easily be seen that cp may be taken such that the

real part and the imaginary part of e i<px are orthogonal vectors. Suppose that x.

:f

0, x

:f

O. Then lIoAII2 is given by (13). We have

~ r or ""T"" T -T .... "'T ... ""T '" "'H-x.x.x x +x x x .. x·. x. x + x .x. x x ~ lor_or r r o~ o~ or or o~ o~ 0 0 T T ;:: -...,T=---·-=T~- = -H-xxx.x. xx'+x.x. xx r r ~ ~ r r ~ ~

Because (see (11» we have

Thereby showing that the real disturbance on A generally is larger than the corresponding complex disturbance. An analogous result can be proven

for (14), (15), (17), (18), (19). Thus we have shown that in order to

measure the distance between (A,B) and UNCO we generally find that case (c) gives a smaller distance than case (r).

4. Special cases

In this section we consider two special cases

(a) only disturbances on A are allowed (oB

=

0)

Case (a) can be handled easily because we only have to restrict x in (10) such that

x~

=

O.

Case (b) forces one to compute the eigenvectors of A. Then the minimal disturbance oB on B~ such that (A, B + oB) is. controllable, is

where v is an eigenvector of A such that

is minimal.

If computation of the eigenvectors of A presents problems one might approximate case (b) by taking d

A and dB in (10) such that dA / dB is "large".

- 5. Discussion

A method to compute the distance between a system and the set of uncon-trollable systems has been described. A disadvantage of this method is that one still needs to minimize a function of 2n variables (where n is

n

the dimension of the system) on the hypersphere in ~ • The extra freedom,

which exists for this minimization problem because e iqlx

~orl7esponds

to a

minimizing-',ector of (10) for any ql € ]R whenever x is such a vector, may ~Q delt with by requiring that the real and imaginary parts of x are orthogonal vectorS!. A vector e iqlx gives rise to the same disturbance

If one considers the case of real disturbances, then a serious difficulty appears because the, to be minimized, object function is not continuous anymore • Therefore computation of the controllability measure using com-plex disturbances seems to be more attractive.

A nice property of a minimal complex disturbance (oA,oB) is that both SA and oB are rank one matrices.

While controllability of a system ~s neither affected by state space

trans-formation nor by feedback we generally have for the controllability measure as described in this paper

d{(A,B),UNCO}

~

d{(TAT-1 ,TB),UNCO}

d{(A,B),UNCO} ~ d{(A+BF,B),UNCO} •

If T is unitary then the controllability measure is the same for (A,B) and -}

(TAT ,TB).

It is easily seen from examples that feedback may enlarge the distance between a system and UNCO but that it also may reduce this distance con-siderably. In order to illustrate this we consider the following situation.

Let (A,B) be a controllable system. Let cr. be the smallest singular value

m~n .

of B. If m < n we may take cr. to be zero because we may add columns m~n

(to B) consisting only of zeroes.

Consider a sequence (~, k = 0,1,2, .•• ) such that

",n ~ € \II II x k II

=

I

~B ~

0 lim

~BBT~

k..;.oo k == 0,1,2, •.. 2

=

a . m~n

Let the sequence of m x n-matrices (F

k, k=O, 1,2, ••• ) be defined by

where (A

k, , k=O, 1,2, ..• ) is a sequence of (complex) numbers. Then

d{(A + BF

k, B),UNGO}

~ X:BBT~

k = 0,1,2, ••.

because

This shows that the distance to UNGO may be reduced considerably using feedback. I f 0 ,

:f:

0 then the minimal distance which can be ob.tained

m~n

in this sense is 02, . If 0 , = 0 (for instance if m < n) then UNGO may

m~n m~n

be approached arbitrarily close using feedback matrices (whose norms tend

.to. infinity).

The codnro11abi1ity measure as described in this paper is not directly related to the singular values of the controllability matrix

[B AB

,

,.

..

,

An-1B] •

In order to show this we consider systems (A ,B ) with dimension n where _{n n}

r

r

0. :,

-·

..

L

.

.

o

~

. . ..

0

.

_.

l

: .

·

_•

... ...

'.,

_{•• " "'..··-1}

.

0

·

.

·

.

·

.

o •••.•••••.•• ' :.

0 for n = 1,2,3,4,5,10,15,20.

o

o

The singular values of the controllability matrix of (A ,B ) all are 1 _{n n} for each n.

However,d{(An,Bn),UNCO} (with d

A

=

dB

=

1) depends on n as is shown Ln the following table n 1 2 3 4 5 10 15 20 d{ (A ,B ) ,_UNCO} n n 1.00 0.75 0.50 0.35 0.25 0.08 0.04 0.02 controllability measures of (A ,B ) n n

This table indicates that large systems tend to be close to UNCO. This holds if one permits disturbances on each element of a system.

Consider also the system

Again the singular values of the controllability matrix all are 1 for any ~.

is not controllable. Therefore (A ,B) is close to UNCO.if a is large. _{a ... ,.}

Often a system has many fixed zeroes and / or ones. The method in this paper does not allow for fixed elements in the A-matrix or the B-matrix. Therefore a different method has to be used in order to compute the controllability measure for such cases.

6. References

[1] Paige, C.c.; Properties of Numerical Algorithms Related to Computing Controllability. IEEE Trans. A.C. Vol. AC-26, no. I, pp. 130-139, 1981.

[2J R ao, C R •• ; M" ~tra, S K . • ; G enera~z 1" ed Inverse of Matrices and its Applications. (Wiley), 1971.